blackAs noted in footnote REF , a recent arXiv revision of [1]} as of this writing proposed a computationally efficient version of the PRECiSE algorithm.
They used a similar idea of lower bounding the multiplicative wealth by an inequality of [2]}.
The difference in our work is twofold.
First, we rely on Lemma REF , which is a higher-order-moments generalization of the lower bound; see Appendix REF .
In the next section,
we also empirically show that the LBUP algorithm with \(n>1\)  clearly improves LBUP with \(n=1\) , which corresponds to the technique of [2]} and thus [1]}.
Second, rather than maximizing over the betting \(b\in [0,1]\)  and invoking the regret bound, we take a mixture over the lower bound as if it were the target wealth process.
While conceptually being simpler in the sense that it does not invoke a regret bound, the downside of our mixture approach is the computational bottleneck in the numerical integration step; see Remark REF .
We finally remark that a mix-and-match approach of our higher-order-moments bound (Lemma REF ) and the regret approach [1]} is natural;
however, we note that maximization of the lower bound over \(b\)  in Lemma REF  may not admit a closed-form expression for \(n>1\) , which necessitates to invoke a numerical optimization. We do not pursue this direction in the current paper.
